Introduction
Regulators have identified four primary methodologies for estimating the market risk premium (MRP) to set regulatory prices First, MRP estimates can be derived from survey evidence, including insights from corporate executives, academics, auditors, and accountants Second, the MRP can be calculated using dividend growth models Third, historical averages of annual excess returns, which compare equity returns to the risk-free rate, can provide MRP estimates Lastly, MRP estimates may be conditioned on current information, taking into account factors such as market volatility, dividend yields, and the risk-free rate.
This paper examines the differences between the third and fourth methods, highlighting the critical issues surrounding the debate on conditional versus historical unconditional estimates of the Market Risk Premium (MRP) This discussion is intricately linked to the predictability of excess returns, stemming from the connection between expected returns and the required rate of return Notably, the annual investment returns report by Dimson, Marsh, and Staunton from Credit Suisse Global reinforces these findings.
In the Investment Returns Sourcebook 2012, the authors explore whether market risk premium (MRP) estimates should be conditional, highlighting the ongoing debate regarding the predictability of returns, which remains unresolved.
Despite extensive research, the debate surrounding predictability remains unresolved A special issue of the Review of Financial Studies features leading scholars presenting conflicting perspectives, notably between Cochrane (2008) and Campbell and Thompson.
(2008) arguing for predictability, whereas Goyal and Welch (2008) find that
Cochrane's (2011) Presidential Address highlights the ongoing debate regarding the effectiveness of certain models in aiding investors who rely solely on publicly available information to time the market profitably This controversy remains significant, as noted by Dimson et al (2012).
Welch and Goyal (2008) assert that investors forecasting excess returns should rely on historical averages rather than current information Their research suggests that estimates of the Market Risk Premium (MRP) are more accurate when based on historical data Similarly, Dimson et al (2012) emphasize that, for practical applications, it is challenging for predictors of equity premiums to surpass long-term historical averages.
In conclusion, the equity premium is expected to change over time due to fluctuations in market volatility and varying levels of investor risk aversion Nevertheless, these effects are likely to be temporary, resulting in brief periods of significantly lower or higher returns.
In efficient and equilibrium markets, expected returns align with the required rate of return, suggesting that both are influenced by current information when returns are predictable While stock prices can affect immediate returns, their impact on long-term performance tends to diminish over time Additionally, volatility typically does not remain at elevated levels for extended periods, and investor sentiment often reverts to the mean Consequently, for forecasting the long-run equity premium, relying on extrapolation from the longest available historical data proves to be the most effective approach.
When they refer to ‘the long run equity premium’, they have in mind forecast horizons of about five years 2
The ongoing debate on the predictability of equity returns, as highlighted by Dimson et al (2012), remains unresolved, with John Cochrane's influential work underscoring the controversy This paper summarizes the evolution of research on predictability since the 1960s, identifying three distinct phases Cochrane (2001) marked the transition from the first phase, which emphasized unpredictability, to a second phase advocating for the predictability of returns However, Ang and Bekaert (2007) noted a shift to a third phase, which calls for skepticism regarding return predictability, particularly over the medium- and long-term A pivotal moment in this discourse was the 2008 issue of the Review of Financial Studies, where Welch and Goyal (2008) significantly challenged the notion of predictability While the second phase supports conditional estimates of the Market Risk Premium (MRP), the third phase questions the validity of these estimates compared to the unconditional historical average.
The third phase of research addresses concerns from regulated businesses regarding the use of a historical average for MRP estimates, which they argue is 'backward-looking.' These businesses question the compatibility of historical excess returns with the forward-looking nature of the Capital Asset Pricing Model (CAPM) However, if we accept the findings of Welch and Goyal (2008), the historical average can actually serve as a forward-looking measure, as it effectively predicts future excess returns and lacks clear superior alternatives.
While Section 2 of the paper identifies the key phases in the general debate about predictability, Section 3 examines specific debates about which explanatory variables can be
Dimson et al (2012) provide evidence supporting the predictability of excess returns over forecast horizons of five years or less, focusing on three key predictor variables: dividend yields, interest rates, and volatility While earlier sections delve into the academic literature on return predictability, the latter part highlights practical challenges regulators may encounter when estimating a conditional Market Risk Premium (MRP) Even if excess returns can be predicted using specific variables, regulators face at least three significant issues in applying these variables for MRP estimation.
Recent literature addressing skepticism about predictability in the third phase of research has explored a variety of increasingly diverse and complex models of returns This complexity poses challenges for regulators contemplating conditional models of the Market Risk Premium (MRP), making it difficult to select and implement an appropriate model amidst the multitude of options available.
The third phase of research highlights significant concerns regarding the stability of excess return models, with various studies indicating that parameter values in these models often fluctuate over time This raises questions about the regulator's ability to determine the Market Risk Premium (MRP) based on a variable that may have an evolving relationship with excess returns.
(3) Apparently significant relationships between variables and excess returns may reflect data-mining.
In conclusion, the ongoing debate among researchers regarding predictability remains unresolved, as noted by Dimson et al (2012, p 36) While the second phase of research may advocate for a conditional estimate of the Market Risk Premium (MRP), the third phase could support an unconditional estimate However, there are three compelling reasons why regulators might prefer to utilize an unconditional estimate of the MRP in practice.
Predictability versus unpredictability: three phases of research
The first phase: the unpredictability of returns and random walks
Cochrane (2005) identifies two distinct phases in asset pricing research, noting that the 'first revolution in finance,' which peaked in the early 1970s, highlighted the near unpredictability of stock returns In contrast, a newer wave of empirical research suggests that stock returns are predictable, particularly over the business cycle and longer time horizons While Cochrane aligns himself with this second phase, he also summarizes key propositions from the first phase of research.
Stock returns exhibit a high degree of unpredictability, resembling random walks, and expected returns remain relatively stable over time Any perceived predictability is often a statistical illusion that disappears in practical applications or fails to account for transaction costs (Cochrane, 2005, p 389).
In the early stages of research, the unpredictability of stock returns was attributed to the concept of efficient markets, which Fama (1970) defines as those where prices fully reflect available information This efficiency complicates return predictions, as highlighted by Samuelson's assertion that if a price were guaranteed to rise, it would have already done so Cornell (1999) further exemplifies this principle, reinforcing the notion that market efficiency leads to random price fluctuations.
Suppose, for example, that someone were to write a convincing book entitled The
The Crash of 2000 suggests that the new millennium will see a significant decline in share prices If the book's arguments were compelling, investors would likely sell their stocks before the millennium began However, if a substantial number of investors acted on these predictions, stock markets would have already declined upon the book's release, indicating that the predictions may not hold true.
3 As noted above, the first edition of Cochrane’s book appeared in 2001.
The central argument posits that if markets operate efficiently, the public release of information will lead to a decrease in prices Consequently, the idea that prices can be predicted creates a contradiction.
In the initial phase of research, theoretical arguments regarding unpredictability gained backing from empirical studies on stock markets Fama (1991) highlights that while short-run correlations—such as those between daily, weekly, and monthly returns—often appeared positive, researchers found insufficient statistical evidence to dismiss the notion of constant expected returns.
1578) summarises the empirical findings of the first phase of research:
Early research on predictability shows limited statistical power, leading to a minimal portion of return variance explained by expected returns Consequently, the hypothesis of market efficiency and stable expected returns is generally regarded as a reliable model.
The second phase: from unpredictability to predictability
In the late 1980s, a new wave of research emerged, challenging the earlier findings on market efficiency Fama (1991, p 1609) summarizes this 'new evidence' on predictability in his survey article, highlighting a shift in understanding market dynamics.
Recent evidence indicates that certain variables, such as Dividend-to-Price (D/P), Earnings-to-Price (E/P), and term-structure variables, provide a more reliable forecast of expected returns over time, particularly following the Great Depression While these variables account for less than 5% of the variance in monthly returns, they explain approximately 25-30% of the variance in returns over 2 to 5 years This suggests that expected returns can experience significant, gradually diminishing fluctuations away from their long-term averages.
Like Fama, Cochrane (2005, p 390) emphasises the contrast between short- and long-horizon predictability He outlines the findings of this ‘new generation’ of empirical evidence as follows:
The dividend/price ratio and term premium are significant predictors of stock return variation, particularly over the business cycle and extended timeframes However, daily, weekly, and monthly stock returns remain largely unpredictable.
The second phase of research builds on earlier findings, recognizing that returns are nearly unpredictable in the short term, while asserting that they become predictable over longer time horizons.
Cochrane (2005) argues that predictability aligns with efficient markets, suggesting that a new understanding of market dynamics does not negate the notion of competitive and efficient markets However, this raises questions about the relationship between market efficiency and unpredictability, as discussed in section 2.1 of this paper The intuitive argument presented by Cornell highlights that if a price increase were certain, it would have already been reflected in the current price This paradox invites further exploration into how the second phase of research can reconcile these seemingly contradictory perspectives on market behavior.
To understand the distinction between normal and abnormal returns, it is essential to recognize that while efficient markets prevent the exploitation of information for abnormal returns, they do not rule out the possibility of predictable changes in normal returns over time Peirson et al (2006) highlight that the random walk hypothesis, which suggests that prices follow a random path and that returns are unpredictable, is often incorrectly linked to market efficiency In reality, the efficient markets hypothesis (EMH) indicates that information cannot be utilized to achieve abnormal returns, but it does not negate the potential for normal returns to exhibit predictability.
The random-walk hypothesis, often linked to market efficiency, does not necessarily support the notion of market efficiency, as a random walk does not imply or is implied by it This model posits that price changes are independent and identically distributed over time; however, these assumptions are not essential for market prices to accurately reflect all available information The Efficient Market Hypothesis (EMH) simply asserts that analyzing prices cannot lead to abnormal returns.
Even in efficient markets, returns may not be entirely unpredictable, as theoretical models suggest that normal returns can evolve in predictable ways over time This article explores three phase-two theoretical models that shed light on this phenomenon The first model indicates that excess returns may be predictable based on dividend yield information The second model offers a similar perspective, linking excess returns to the risk-free rate Lastly, the third model examines the relationship between excess returns and return volatility, providing a comprehensive theoretical framework for understanding return predictability.
Lettau and Ludvigson (2001) highlight a growing consensus on the predictability of excess returns, emphasizing the significance of various financial ratios They identify price-dividend ratios, price-earnings ratios, and dividend-earnings ratios as key predictors in assessing investment performance.
The predictability of stock returns is well-established, with financial metrics like price-to-dividend, price-to-earnings, and dividend-to-earnings ratios demonstrating significant predictive capabilities for excess returns beyond Treasury-bill rates.
Over a decade ago, Lettau and Ludvigson proposed that the predictability of returns was uncontroversial However, since the publication of their article, a new phase of research has emerged, reigniting the debate over the predictability of returns.
The third phase: a renewed ‘healthy skepticism’ about predictability
In the July 2008 issue of The Review of Financial Studies, a significant shift in research is highlighted, marking the emergence of a third phase that contrasts with the previous phase This is exemplified by the lead article by Welch and Goyal, titled ‘A Comprehensive Look at the Empirical Performance of Equity Premium Prediction.’ Utilizing US data up to 2005, the authors assess the predictability of excess returns by examining various variables, such as the dividend yield, stock variance, and the risk-free rate (Treasury-bill rate) Their findings indicate that these models for predicting excess returns have largely underperformed.
Welch and Goyal differentiate between 'in sample' (IS) and 'out of sample' (OOS) performance in forecasting models, highlighting the significance of OOS forecast performance as emphasized by Brooks (2008, p 245).
In-sample forecasts are produced using the same data that estimated the model's parameters, leading to generally accurate predictions To effectively evaluate a model's forecast accuracy, it's essential to avoid using all observations for parameter estimation Instead, retaining a portion of the data, known as the holdout sample, allows for the creation of out-of-sample forecasts, providing a more reliable assessment of the model's performance.
Welch and Goyal (2008) discovered that many variables previously believed to predict excess returns yielded unsatisfactory in-sample (IS) forecasts Additionally, they noted that most variables that showed strong performance in IS analysis did not translate well to out-of-sample (OOS) predictions.
Most financial models have lost their significance in sample (IS) testing, and those that remain often fail basic regression diagnostics Additionally, many models exhibit poor out-of-sample (OOS) performance, particularly in predicting outcomes late in the sample period While it is possible to find and justify some statistically significant models, the overall results indicate a need for skepticism regarding predictions of the equity premium, especially as of early 2006 The lack of robustness in these models raises concerns about their reliability.
This ‘healthy skepticism’ about predictability characterises the third phase of research on stock returns When assessing the forecasting power of various variables, Welch and Goyal
In their 2008 study, researchers compared the forecasting performance of various variables against forecasts derived from a simple historical average The findings revealed that the simple historical average performed equally well, if not better, than the diverse variables analyzed in the study.
OOS, most models not only fail to beat the unconditional benchmark (the prevailing mean) in a statistically or economically significant manner, but underperform it outright’ (Welch and Goyal, 2008, p 1504).
The debate over whether to measure the Market Risk Premium (MRP) using an unconditional historical average or a conditional approach is significant for regulators Critics argue that employing a historical average in the forward-looking Capital Asset Pricing Model (CAPM) framework is inconsistent However, Welch and Goyal (2008) suggest that the historical average can be viewed as a forward-looking risk premium measure, as it forecasts future excess returns effectively Their findings indicate that the unconditional historical average performs comparably to other conditional estimates, reinforcing its validity as a predictor of future excess returns.
Welch and Goyal (2008) is not the only article in the June 2008 edition of Review of
In the realm of financial studies, the findings from phase two regarding predictability are being scrutinized Boudoukh et al (2008) echo this skepticism in their article, "The Myth of Long-Horizon Predictability," challenging the assertion that long-horizon predictability significantly outperforms short-horizon predictability.
Welch and Goyal (2008) highlight significant concerns regarding phase-two research, particularly emphasizing the necessity of out-of-sample (OOS) tests and the instability of proposed equity return models, a topic further explored in Section 4.2 Additionally, Bossaerts and Hillion (1999) note issues related to data mining, revealing that when employing formal model selection criteria to mitigate this problem, models predicting excess returns tend to underperform in out-of-sample evaluations.
The analysis reveals a lack of reliable out-of-sample predictability, with the exception of Japan Notably, the t-ratios for the slope coefficient in the Seemingly Unrelated Regression (SUR) of out-of-sample outcomes against predictions are not statistically significant As a result, we conclude that models selected using formal criteria do not demonstrate external validity.
The Bossaerts and Hillion (1999) paper will be discussed again in Section 4.3 below, which elaborates on the problems arising from data mining.
Phase-two research faces criticism due to technical statistical challenges in testing and estimating equity return models Common explanatory variables, such as dividend yields, book-to-market values, and interest rates, exhibit high persistence Zhu (2013) highlights that in these situations, ordinary least squares (OLS) estimation may lead to 'spurious regressions,' which can distort the results.
‘finite sample bias’ 4 In such a case, standard tests of significance are invalid (Zhu, 2013, p.
194) To solve this problem, Zhu constructs an alternative method of estimation – his
The 'jackknife' technique, as proposed by Zhu, offers superior estimates compared to traditional OLS estimates To highlight this difference, Zhu analyzes excess returns in relation to dividend yields and short-term interest rates He discovers that while OLS estimates suggest a strong predictability of returns, this predictability disappears entirely when employing his preferred jackknife method.
The strong statistical evidence supporting predictability diminishes significantly when finite-sample biases are accounted for, suggesting that these biases largely account for the observed predictability Consequently, these findings raise concerns about the validity of earlier studies that claimed predictive power from the dividend yield and short rate (Zhu 2013, p 211).
Torous et al (2004) highlight the significant technical statistical challenges posed by the high persistence of explanatory variables in equity return models They propose an estimation technique to overcome these challenges, yielding results that differ markedly from those found in phase-two studies, similar to the findings of Zhu (2013).
The study reveals minimal evidence of predictability for investment returns beyond a one-year horizon within the 1926-1994 sample period, including both pre-1952 and post-1952 subsamples However, in the post-1952 period, several explanatory variables, such as dividend yield, term spread, and short-term interest rates, effectively forecast returns within a shorter timeframe of less than 12 months This finding challenges previous research, indicating that predictability is more reliable at short horizons rather than long ones.
Other phase-three research has similarly questioned the predictability of returns over longer horizons For example, Ang and Bekaert (2007, p 696) find that:
Our results suggest that predictability is mainly a short-horizon, not a long- horizon, phenomenon
In the context of an ACCC/AER regulatory decision, short-horizon predictability – predictability over horizons of a year or less – is less relevant than predictability over longer
4 For another paper that emphasises the ‘spurious regression’ problem, see Ferson et al (2003).
5 For another technique designed to solve this problem, see Stambaugh (1999).
The effect on the MRP of the dividend yield, risk-free rate and volatility
Dividend yields and the MRP
The second phase of research on predictability highlights the significant relationship between the market risk premium (MRP) and dividend yield, suggesting that a lower dividend yield forecasts lower excess returns Campbell and Cochrane (1999) have explored this connection, with their theoretical insights summarized by Cochrane (2005).
The predictability of returns based on price/dividend ratios can be attributed to changing risk aversion levels among investors; during economic booms, individuals tend to become less risk averse as their consumption and wealth rise, whereas in recessions, they exhibit increased risk aversion as both consumption and wealth decline.
Cochrane argues that during economic booms, dividend yields are typically low due to high stock prices, while investor risk aversion decreases as consumption and wealth rise This shift leads investors to demand a lower risk premium for the same level of risk, resulting in a positive correlation between dividend yields and the risk premium.
According to Cochrane (2005), risk aversion is influenced not by the absolute level of consumption, but by how current consumption compares to recent trends, reflecting an individual's accustomed standard of living.
Risk aversion cannot be directly linked to consumption and wealth levels, as these tend to rise over time while equity premiums remain stable To explore this idea, we need a model where risk aversion is influenced by consumption or wealth relative to a historical trend People may become accustomed to a certain standard of living, making a decline in consumption feel more painful after experiencing prosperity, even if the same level of consumption would have been viewed positively following difficult times This perspective helps clarify why recessions are perceived as severe events, despite a recession year potentially ranking as one of the best in human history (Cochrane 2005, p 467).
In summary, a low dividend yield often correlates with elevated stock prices, typically seen during economic booms During these periods, risk aversion diminishes, contributing to a lower market risk premium (MRP).
In the second phase of predictability research, several empirical studies highlighted a positive correlation between dividend/price ratios and future returns Fama and French's seminal 1988 paper established a crucial distinction between short-term and long-term returns, asserting that while dividend yields do not account for returns in the short term, they significantly influence returns over extended periods.
Our tests reinforce existing findings that the predictable component of returns accounts for only a small portion of short-horizon return variances, with regressions on yields explaining less than 5% of monthly or quarterly variances Notably, our results enhance the statistical evidence that the predictable component plays a more significant role in long-horizon returns, as regressions on dividend-to-price ratios (D/P) often elucidate over 25% of the variances in two- to four-year returns.
Fama and French (1988) demonstrate that equity returns can be forecasted based on dividend yields, while their 1989 study reveals that excess returns also show predictability from the same yields Consequently, they conclude that dividend yields are a significant predictor of both equity and excess returns.
The findings indicate that excess returns closely mirror the patterns observed in real returns Consequently, the fluctuations in anticipated real stock returns, as indicated by dividend yields, are also reflected in the expected premiums of stock returns compared to one-month bill returns.
Cochrane (2005) aligns with the findings of Fama and French (1988, 1989) in the second phase of research on predictability By analyzing data from 1947 to 1996, Cochrane (2005, p 392) presents results that demonstrate the relationship between excess returns and the dividend price ratio.
The one-year horizon 0.15 R 2 is not particularly remarkable However, at longer and longer horizons larger and larger fractions on return variation are
Cochrane (2005) explains that while dividend yields may not effectively predict short-term returns, they can serve as a strong indicator of long-term performance due to their nature as a "slow-moving variable." This means that even slight predictability in daily returns accumulates over extended periods, much like how one can forecast a gradual increase in temperature during spring Over a five-year horizon, approximately 60% of the variation in stock returns can be anticipated using the price/dividend ratio, demonstrating the value of long-term forecasting in financial markets.
Lettau and Ludvigson (2001) highlight the significant difference between long and short horizon forecasts of excess returns based on dividend yields, reinforcing findings from the second phase of research discussed in section 2.2.
Research indicates that the dividend yield has minimal predictive power for returns over a one-year horizon, with its coefficient estimate showing no significant difference from zero However, it emerges as a significant predictor for returns at a six-year horizon This aligns with previous findings, suggesting that while the dividend yield effectively forecasts long-term returns, it lacks the ability to predict short- and medium-term returns.
The third phase of research challenges the conclusion that dividend yield serves as a reliable predictor of long-term returns This skepticism is part of a broader critique on the predictability of returns, particularly over extended periods, as highlighted by studies from Welch and Goyal (2008), Boudoukh et al (2008), Ang and Bekaert (2007), and Torous et al.
(2004) and Zhu (2013)) express, in particular, concerns about the use of the dividend yield as a predictor.
The risk-free rate and the MRP
The discussion in Section 3.1 highlights the relationship between the Market Risk Premium (MRP) and dividend yield, suggesting a broader understanding of MRP fluctuations over time It explains the expectation that the MRP is counter-cyclical, increasing during economic slowdowns and decreasing during periods of economic boom Fama and French (1989) support this notion of counter-cyclical MRP behavior, as does Cochrane.
Expected returns fluctuate with the business cycle, requiring a higher risk premium for investors to hold stocks during economic downturns (2005, p 392) Cochrane notes that both the market risk premium (MRP) and dividend yield exhibit systematic movements alongside the business cycle, indicating a correlation between the two McKenzie and Partington (2013, pp 25-26) explore the potential for a countercyclical MRP to create a negative relationship with the risk-free rate, suggesting that if the MRP is counter-cyclical and the risk-free rate is pro-cyclical, they would be negatively correlated This theoretical framework presents a foundational argument for the observed negative correlation between the risk-free rate and the MRP, supported by additional perspectives (Breen et al., 1989, p 1178-9).
Recent empirical studies in the United States have identified a negative correlation between the risk-free rate and future excess returns Notably, Lettau and Ludvigson (2001) demonstrated this negative relationship between the 'relative bill rate' and anticipated excess returns, a finding supported by the work of Ang and Bekaert.
(2007) show that three-month Treasury bills have a negative relationship with future excess returns.
When assessing the regulatory cost of capital, it is crucial to differentiate between short-term and long-term forecasts Research indicates that the interest rate is a reliable predictor primarily for short forecast horizons Ang and Bekhaert (2007) assert that while the short rate is the most robust variable for predicting future excess returns, its significance diminishes for longer periods, particularly beyond five years Similarly, Lettau and Ludvigson (2001) reveal that the relative bill rate significantly correlates with future excess returns only for horizons of one year or less.
When determining the regulatory cost of capital for a five-year period, regulators must focus on returns throughout the entire duration rather than just the initial year Research by Ang and Bekhaert (2007) and Lettau and Ludvigson (2001) indicates that the risk-free rate is only relevant for excess returns over shorter forecast horizons, making it unsuitable for setting the Market Risk Premium (MRP) in a five-year context Despite this, some consultants for regulated businesses have incorrectly used Lettau and Ludvigson's findings to support a negative correlation between the risk-free rate and the MRP, overlooking the broader implications for the regulatory period.
The relative bill rate is a detrended measure of the bill rate, as highlighted by Campbell (1991, p 166), who emphasizes the necessity of using a detrended rate due to the potential nonstationarity of short-term interest rates during the sample period He notes that a simple method for achieving this is by subtracting a one-year moving average, although it is considered a rudimentary approach.
11 See, for example, Competition Economics Group, Internal Consistency of Risk Free Rate and MRP in the
The market risk premium is generally observed to move inversely to the risk-free rate, particularly during significant fluctuations in the latter Research by Lettau and Ludvigson highlights a statistically significant negative correlation between de-trended government bond rates and changes in log excess returns, akin to the market risk premium However, it is important to note that this correlation is only significant over shorter time horizons than the regulatory period, limiting its relevance for estimating the regulatory cost of capital.
In his 2013 study, Zhu discovered that the risk-free rate is an inadequate predictor of excess returns over short time frames By utilizing a jackknife technique to address short-term bias, Zhu found that any evidence supporting the predictive power of the risk-free rate disappears entirely.
Volatility and the MRP
The relationship between the Market Risk Premium (MRP) and volatility is rooted in Merton's intertemporal Capital Asset Pricing Model (ICAPM) from 1973 In this model, the MRP is dynamic and varies over time Armitage (2005) describes the ICAPM as a multifactor model where an asset's risk premium is influenced by its sensitivity to various state variables.
Merton (1973) is often referenced in discussions regarding the relationship between the market risk premium (MRP) and volatility; however, he does not explicitly analyze this connection in his initial paper Instead, Merton (1980) delves into this relationship, building on and examining specific cases from his earlier work He posits that, under certain assumptions, the MRP at any given moment is positively correlated with the conditional volatility at that same time.
The coefficient θ depends, in turn, upon investors’ coefficients of relative risk aversion 12
The relationship between the Market Risk Premium (MRP) and volatility appears intuitively logical, as higher risk typically warrants a greater premium for investors However, empirical data does not consistently support this correlation Consequently, some researchers argue that the perceived positive link between MRP and volatility may not be as theoretically sound as it first appears, with Cornell (1999) highlighting this skepticism.
It turns out that the economic intuition that periods of high price variability should also be characterized by high stock-market returns is false.
12 See Merton (1980, pp 329-330) for this result For examples of studies which rely on Merton (1973,
1980) to derive a relationship between expected returns and conditional volatility, see Poterba and Summers
(1986), p 1143; Harvey (1989), p 291; Dean and Faff (2001), pp 172-73; Glosten, Jagannathan and Runkle
Cornell suggests, furthermore, that the ambiguity of the relationship between returns and volatility ‘is not so surprising’ in the light of the following observation of Glosten et al.
Initially, it seems that risk-averse investors would demand a higher risk premium during riskier periods However, this may not be the case, as riskier times can align with investors' increased capacity to manage certain risks Additionally, during uncertain future periods, investors might choose to save more, which could also lessen the need for a larger risk premium In scenarios where all available productive assets are risky and no risk-free options exist, the demand for risky assets may increase significantly, leading to a reduced risk premium Consequently, both positive and negative correlations between the conditional mean and conditional variance of excess stock returns can be theoretically justified.
The relationship between market risk premium (MRP) and volatility, while seemingly plausible, remains theoretically ambiguous According to Glosten et al (1993), reliance on theory alone does not adequately support a positive correlation between MRP and volatility Therefore, empirical evidence is essential to clarify the dynamics between these two financial concepts.
Cornell (1999) highlights the ambiguity in the theoretical relationship and notes that empirical evidence is inconsistent While some econometric studies report an insignificant relationship, others that do find significance vary in their conclusions regarding the direction of this relationship.
The literature on stock returns and return variability features notable contributions from researchers such as Black (1976), Merton (1980), and French et al (1987), highlighting that return variability fluctuates over time However, there is a lack of consensus among these studies regarding its relationship with the risk premium; some suggest a positive correlation, others indicate a negative one, while some find no significant relationship Ultimately, the evidence points to a remarkably weak connection between stock returns and return variability, suggesting that return variability is not an effective variable for modeling changes in the risk premium.
A number of other researchers echo Cornell’s conclusion that the empirical literature on the relationship between volatility and the equity premium is inconclusive Thus Scruggs (1998, p 575) observes that
Despite theoretical predictions suggesting a positive correlation between the market risk premium and conditional market variance due to investor risk aversion, empirical studies have yet to reach a consensus on the nature of this relationship.
Scruggs offers a valuable overview of empirical research findings, highlighting the varied results regarding the connection between risk premium and variance The summarized data, sourced from Scruggs (1998, p 577), illustrates the differing empirical outcomes on this relationship.
Survey of Empirical Research on the Relation between the Risk Premium and Volatility 13
Paper Empirical relation between risk premium and market variance
French, Schwert and Stambaugh (1987) Insignificant positive
Turner, Startz and Nelson (1989) Significant negative
Significant positive Significant positive Baillie and DeGennaro (1990) Significant positive
Insignificant positive Insignificant positive Insignificant positive Glosten, Jagannathan and Runkle (1993) Insignificant positive
In another summary of the empirical literature, Whitelaw (1994, p 515-516) also emphasises the variety of divergent findings about the relationship between the MRP and volatility:
Market intuition indicates a positive relationship between risk and returns, leading researchers to explore the connection between expected returns and conditional volatility However, previous empirical studies examining the correlation between these two aspects of stock market returns have produced inconsistent findings.
13 Some papers provide multiple estimates because they may use multiple proxies, data sets, models of variance, estimation methodologies or specifications of the risk-return relation.
Dean and Faff (2001, p 169-170) provide a similar characterisation of the empirical literature on the relationship between expected returns and volatility:
Financial theory suggests a positive correlation between expected returns and variance; however, researchers struggle to reach a consensus on the direction of this relationship and face challenges in predicting its magnitude.
Recent studies highlight the inconclusive nature of empirical research on the link between volatility and expected returns Armitage (2005) supports Cornell's assertion that the connection between stock returns and return variability is notably weak Similarly, Bollerslev et al (2009) characterize the existing empirical literature in a comparable manner, underscoring the ongoing debate in this area.
The classical intertemporal Capital Asset Pricing Model (CAPM) developed by Merton in 1973 underpins the traditional risk-return tradeoff observed in aggregate market returns However, despite extensive empirical studies aimed at estimating this risk premium, finding a consistent, time-invariant relationship between expected return and volatility has remained largely elusive.
When Bollerslev et al (2009) talk of ‘a significant time-invariant expected return-volatility tradeoff’ they mean a significant time-invariant positive relationship between expected returns and volatility.
Based on the observations of various researchers, including Cornell (1999) and Scruggs (1998), there is no consensus in the empirical literature regarding a strong positive relationship between the Market Risk Premium (MRP) and volatility In light of this conclusion, some researchers, such as Guo and Savickas, have sought to develop more complex models of expected returns that incorporate different measures of volatility to better understand return dynamics.
Recent studies by Bollerslev et al (2006) and Bollerslev et al (2009) highlight a significant shift from the traditional view of the relationship between volatility and the Market Risk Premium (MRP) typically submitted by regulated businesses As discussed in Section 4.1, the complexity and variety of conditional MRP models in contemporary academic literature raise challenges for regulators in making evidence-based selections of specific models and effectively implementing them.
Reasons for regulators to have practical concerns about conditional estimates
The diversity and complexity of recent models of return predictability
The current research literature on return predictability is characterised by a diversity of distinct and complex models of excess returns.
The third phase of research literature critically reassesses the predictability claims made during the second phase, with notable contributions from researchers like Welch and Goyal (2008).
14 This observation is a commonplace of the academic literature, and is articulated by Peseran and Timmermann
(1995, pp 1201-2) in the passage below They note that there is an interpretation of return predictability – their
‘second interpretation’ – according to which markets are inefficient, and the MRP is constant despite the predictability of excess returns:
Recent studies suggest that stock returns can be predicted using publicly available information; however, the economic implications of these findings are debated One interpretation posits that predictable components in stock returns indicate time-varying expected returns, which could align with an efficient market Conversely, another view considers expected returns to be relatively constant, interpreting the predictability of stock returns as a sign of market inefficiency Ultimately, any theoretical interpretation of excess return predictability is contingent on the model used, leading to inconclusive results.
Peseran and Timmermann’s quotation refers to Fama (1991, p 1577), which makes the same point:
Recent research indicates that past returns, dividend yields, and various term-structure variables can predict future returns However, this finding raises the joint-hypothesis problem, questioning whether return predictability stems from rational changes in expected returns, irrational price deviations from fundamental value, or a combination of both While some researchers express skepticism regarding the predictability of returns, others are exploring more complex models to better understand this phenomenon.
Recent studies highlight the significant impact of phase-three research on contemporary academic literature, showcasing the complexity and diversity of the models developed in response to this research.
1 Rapach et al (2010) open by recognising the current scepticism about return predictability, citing phrase-three research, including Bossaerts and Hillion (1999), Goyal and Welch (2003) and Welch and Goyal (2008) Their response to the problems identified by phase-three research is to move away from ‘individual forecasts’ – forecasts based on a single observed variable Instead, they use fifteen different variables to estimate fifteen individual forecasts. They recommend using a forecast based on a combination of these fifteen individual forecasts.
2 Timmermann (2008) similarly motivates his paper by pointing to the scepticism about predictability articulated in Bossaerts and Hillion (1999), Goyal and Welch (2003) and Pesaran and Timmermann (1995) He suggests that nevertheless there may be ‘local predictability: ‘most of the time stock returns are not predictable, but there appear to be pockets in time where there is modest evidence of local predictability’ (Timmermann, 2008, p 17) If an econometrician runs a range of models, he or she may be able to make local predictions if it is possible to measure which models are working well at different points of time: the econometrician must obtain ‘some indication of when different models produce valuable forecasts and when they fail to do so e.g in the form of a real-time monitoring system tracking how reliable the forecasts have been over the recent time’ (Timmermann,
3 Cooper and Priestley (2009) also begin by acknowledging the skepticism about return predictability that arises from phase-three research, citing the papers by Bossaerts and Hillion
(1999) and Goyal and Welch (2003) They respond by suggesting a novel variable for predicting excess returns − the output gap (the deviation of industrial production from trend)
4 Like the previous three studies, Pettenuzzo et al (2012) opens by pointing to the phase- three research of Bossaerts, Hillion, Goyal and Welch They suggest that the performance of forecasts can be improved by introducing constraints on the forecasting models – in particular, the conditional mean of the equity premium is constrained to be non-negative, and the conditional Sharpe ratio is constrained to lie within certain bounds.
5 Bollerslev et al (2009) examine the relationship between volatility and the risk premium. They open their discussion by pointing to the failure of the empirical literature to establish such a relation, concluding that ‘the search for a significant time-invariant expected return- volatility tradeoff type relationship has largely proved elusive’ (Bollerslev et al 2009, p.
The authors explore the intricate relationship between expected returns and volatility by examining the disparity between two volatility measures Their key finding reveals that the difference between "model-free" implied and realized variances significantly accounts for a notable portion of the variation in quarterly stock returns during the 1990-2007 period.
6 Like Bollerslev et al (2009), Guo and Savickas (2006, p 43) begin by pointing to empirical research which ‘failed to uncover a positive risk-return relation in the stock market across time’ Their response is to model expected returns not as a function of a single measure of volatility, but rather as a function of two distinct measures of volatility, idiosyncratic volatility and stock market volatility They find that whereas these two measures of volatility ‘individually have negligible forecasting power in the in-sample regression, they jointly provide a significant predictor of excess stock market returns’ (Guo and Savickas, 2006, p 43).
Since 2006, six key papers have emerged, showcasing recent efforts to model excess returns amid skepticism regarding predictability in phase-three literature These studies introduce innovative and often intricate model specifications, reflecting a wide array of approaches to excess return modeling in academia However, there remains a lack of consensus on the best methodologies for predicting future excess returns Consequently, if a regulator seeks to establish a time-varying, conditional model of the Market Risk Premium (MRP), selecting an evidence-based approach from this diverse and complex body of literature poses significant challenges, complicating both the selection and implementation of a suitable forecasting model.
Instability in models of return predictability
A number of studies have found instability in models of return predictability – that is, the models tend to change over time.
A model parameter is considered unstable if it varies over time, which has become a key focus in the third phase of research on return predictability, particularly concerning excess returns This instability raises concerns for regulators regarding the use of conditional models for the Market Risk Premium (MRP), as relying on a variable that changes unpredictably complicates accurate adjustments to the MRP Consequently, it becomes challenging for regulators to determine the appropriate MRP modifications in response to fluctuations in the variable This significant issue was briefly mentioned in previous sections, specifically in the analyses of Welch and Goyal (2008) and Timmerman (2008), highlighting the need for a more comprehensive examination.
In Goyal and Welch (2003, p 653), the diagnosis for the poor out-of-sample forecasting performance of dividend yields is that the underlying relationships are unstable:
The primary source of poor predictive ability is parameter instability The estimated dividend-price ratio autoregression coefficient has increased from about0.4 in 1945 to about 0.9 in 2002.
Bossaerts and Hillion (1999, p 407) similarly suggest that model instability accounts for the poor OOS predictability of excess returns:
The poor external validity of the prediction models that formal model selection criteria chose indicates model nonstationarity: the parameters of the ‘best’ prediction model change over time.
The following passage provides more detail about the changes in the model of excess returns over time:
The comparison of regression results between Period I (entire sample) and Period II (1/70-8/80) reveals model nonstationarity, characterized by consistent signs of regression coefficients but significant variations in their magnitudes, particularly with Period II showing the highest values Additionally, Pesaran and Timmerman (1995) highlight an increase in the predictability of U.S stock returns during the 1970s, as noted by Bossaerts and Hillion (1999, p.412).
Bossaerts and Hillion cite the results reported in Pesaran and Timmerman (1995, p 1225), which also emphasise the instability in models of excess returns:
The predictability of stock returns in the U.S has experienced significant changes over time, indicating a lack of a robust forecasting model Key macroeconomic events, such as the 1974 oil price shock and alterations in the Federal Reserve's operating procedures, appear to influence the timing of the inclusion of various regressors in these forecasting models.
Even Lettau and Ludvigson (2001, p 844), a study cited by the regulated businesses in support of return predictability, emphasises the changes in the performance of such models over time:
While our research demonstrates notable out-of-sample predictability compared to other studies, we emphasize that these results do not guarantee forecastability in every situation The past five years have exhibited exceptionally atypical stock market trends, with prices soaring to unprecedented levels when assessed against any reasonable benchmark.
Model instability has emerged as a significant concern in phase-three research over the last ten years Notably, studies by Pesaran and Timmermann (2002) and Paye and Timmermann (2006) highlight the difficulties in predicting returns due to this instability.
Data mining
Findings of predictability may reflect data mining rather than genuine relationships between variables and future excess returns Data mining is a particular problem for research on return predictability.
Data mining, also known as data dredging or data snooping, can occur both intentionally and unintentionally An example of unintentional data mining involves multiple econometricians analyzing the determinants of a variable Y, each testing its relationship with different variables (X1 to X20) Even if Y is not actually related to any of these variables, statistical analysis may yield a false positive, leading one econometrician to report a significant finding while the others do not This situation is exacerbated by academic journals favoring publications that claim to discover significant relationships, resulting in a misleading body of literature on Y's determinants Overall, data mining highlights the risks associated with multiple analyses of the same dataset, potentially distorting research outcomes.
Intentional data mining occurs when an econometrician deliberately seeks to establish a false relationship between variables X and Y, despite the absence of any actual correlation This process involves testing numerous model specifications—such as varying start and end dates, adjusting frequencies, using different proxies, accounting for outliers, and selecting various functional forms and additional variables—until a statistically significant relationship is found This practice highlights the potential for manipulation in data analysis.
Data mining can compromise the validity of statistical significance findings For instance, when a dependent variable Y is analyzed in relation to an independent variable X, a result may indicate that the relationship is statistically significant at the 5 percent level This typically suggests that there is only a 5 percent chance that no relationship exists between the two variables However, if the econometrician arrives at this conclusion only after conducting multiple regressions, the reliability of the finding may be questioned.
In examining twenty different variables, the model only highlights those deemed 'statistically significant.' However, this assertion of 'statistical significance' is questionable, as it cannot be confidently stated that the likelihood of no relationship between the two variables is less than 5 percent, as noted by Verbeek (2008, pp 58-59).
Data snooping occurs when a dataset is utilized multiple times to select model specifications and test hypotheses For instance, if you have 20 potential regressors and evaluate each one, it is highly probable that you will identify at least one as significant, despite the absence of any genuine relationship between the regressors and the dependent variable.
Researchers in financial economics have long recognised that data mining is a particular problem for studies on return predictability Thus in his article on market efficiency, Fama
We should also acknowledge that the apparent predictability of returns may be spurious, the result of data-dredging and chance sample-specific conditions.
He goes on to explain why data mining may be especially problematical in empirical research on return predictability:
The issue of inference is complicated by a pervasive problem of data dredging within the industry Numerous skilled researchers, regardless of their stance on market efficiency, are actively searching for forecasting variables This often leads to the discovery of seemingly "reliable" return predictability that is, in reality, spurious (Fama, 1991, p 1585).
In their article "Data-Snooping Biases in Tests of Financial Asset Pricing Models," Lo and MacKinlay (1990, p 432) highlight the issue of data mining, particularly in the context of return predictability research They emphasize that the prevalence of numerous published studies utilizing the same data set significantly contributes to this problem.
As the number of studies conducted on a specific data set increases, the likelihood of uncovering spurious patterns also rises This phenomenon is particularly evident in the realm of stock market prices, which are among the most extensively analyzed economic metrics Consequently, financial asset pricing models may be especially vulnerable to biases due to the intense scrutiny of stock market data.
The article contends that data mining undermines the validity of traditional significance tests (p 434) Sullivan, Timmermann, and White (1999) explore how data mining can create misleading trading rules that seem to reveal significant predictors of future excess returns, despite lacking genuine predictive power.
Over time, certain trading rules may yield superior performance purely by chance, even within extensive datasets This phenomenon occurs despite the absence of genuine predictive power regarding asset returns, as highlighted by Sullivan, Timmermann, and White (1999).
The challenge of data mining poses significant difficulties for regulators assessing econometric studies that aim to establish a conditional Market Risk Premium (MRP) As a result, regulators may struggle to make informed, evidence-based choices regarding the appropriate conditional MRP model.
Conclusion
This paper examines methods for estimating the Market Risk Premium (MRP) for regulatory pricing, emphasizing a comparison between conditional estimates and historical averages It highlights the relevance of this issue to the ongoing debate regarding the predictability of excess returns and reviews three phases of literature on this topic The findings suggest that the third phase may support the use of historical averages as a reliable estimate of the MRP, indicating that when forecasting excess returns, historical averages often serve as an effective forward-looking estimate.
The ongoing debate among researchers regarding the predictability of excess returns remains unresolved, as noted by Dimson et al (2012) Even if it is acknowledged that excess returns can be somewhat predicted using certain variables, regulators encounter at least three significant challenges in utilizing these variables to accurately estimate a conditional Market Risk Premium (MRP).
Recent literature addressing skepticism about predictability in the third phase of research has led to the development of a diverse and complex array of models For regulators contemplating conditional models of the Market Risk Premium (MRP), selecting the appropriate model based on evidence is challenging due to the variety and increasing complexity of these models, making implementation equally difficult.
The third phase of research highlights significant concerns regarding the stability of excess return models, as numerous studies indicate that parameter values within these models fluctuate over time This instability raises questions about how regulators can determine the Market Risk Premium (MRP) based on specific variables, particularly in assessing the appropriate adjustments needed in response to changes in those variables.
(3) Apparently significant relationships between variables and excess returns may reflect data mining.
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